Period-doubling Bifurcations Research Articles

Dynamics of pumping effect in 1 pipe-2 tanks flow configuration with periodic excitation

This paper is dedicated to analyzing the pumping effect in rigid 1 pipe-2 tanks flow configuration proposed by G. Propst. Our main purpose is to study the dynamic behaviors of the periodic solutions of the governing differential equation, as the periodic solution will result in the periodically forced pump in the configuration. We establish a sufficient condition for the existence of the harmonic solution by applying the topological degree theory. Additionally, we utilize the regular perturbation method and averaging method to examine the stability of the harmonic solution, as well as the existence and stability of the second-order subharmonic solution of the equation. Through numerical bifurcation analysis, we present the bifurcation diagrams of the periodic solutions, which reveal that higher-order subharmonic solutions with varying stability can be generated through period-doubling bifurcation of the periodic solutions.

Finite time SF bifurcation and stability analysis for stochastic time-varying delay system

In this paper, it is found that the systems alternate between oscillatory and steady states in studying the stability of a class of stochastic dynamical systems with fast time-varying periodic delays. We propose a new definition of bifurcation, State Flip (SF) bifurcation, by using the abrupt change in the dwell time ratio of two states. Taking the amplitude of the time-varying delay function δ as the bifurcation parameter, we investigate the dynamics of the evolution of a time-varying delay system driven by Gaussian white noise. Firstly, we convert systems with fast time-varying periodic delays into comparative systems with distributed delays, which in turn are converted into stochastic systems with constant delays for discussion. In order to reduce the stochastic delay equation to the average Itô equation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. We then find that as the amplitude increases from weak to strong, the stochastic time-varying delay system changes from a two-period oscillation to alternating between oscillation and steady, and finally to a steady state. Lastly, we perform numerical simulations of the original stochastic time-varying delay system and the transformed two-dimensional stochastic constant time-delay system respectively to verify that the ideas and methods in this paper are reasonable and effective.

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

The complex dynamics of a predator-prey system in discrete time are studied. In this system, we consider the prey’s Gompertz growth and the square-root functional response. The existence of fixed points and stability are examined. Using the center manifold and bifurcation theory, we found that the system undergoes transcritical bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation. In addition, numerical examples are presented to illustrate the consistency of the analytical findings. The bifurcation diagrams show that the positive fixed point is stable if the death rate of the predator is greater than a threshold value. Biologically, it means that to prevent the predator population from growing uncontrollably and stability of the positive fixed point, the predator’s death rate should be greater than the threshold value.

Complex pattern dynamics and synchronization in a coupled spatiotemporal plankton system with zooplankton vertical migration

In aquatic environments, zooplankton often exhibits life habit of vertical migration and the phytoplankton-zooplankton predator-prey systems in different water layers are therefore coupled together. To uncover the nonlinear mechanisms of pattern dynamics in such coupled plankton system, a spatiotemporal model is developed in this research with the consideration of plankton horizontal diffusion and zooplankton vertical migration. Equation of dispersion curve for the coupled system is derived, and an extension of regular Turing instability is disclosed by determining the number of peaks on the dispersion curve. We find a mechanism named as “projection of characteristics” that local or global characteristics projecting between different subsystems causes the self-organization of complex patterns with multi-scale structures. The combination of various bifurcations with Turing instability controls the system stability and can lead to more complex pattern formation. Notably, when flip bifurcation takes place, the coupled system generates an interesting result that the phytoplankton populations in some water layers become extinct but the zooplankton populations in these layers can still be persistent due to the effect of population compensation. It is also found that the increase of zooplankton migration rate can result to spatiotemporal synchronization of pattern evolution, and that particularly, the pattern dynamics in just a part of subsystems achieve local synchronization. The methods and results presented herein may lead to a better understanding on the emergence of complex plankton patchiness in nature.

Period-doubling bifurcation to chaos in a pure-sextic soliton fiber laser

We numerically investigate the dynamics of period-doubling bifurcation to chaos in a pure-sextic soliton (PSS) fiber laser dominated by sixth-order dispersion (SOD). By properly setting the cavity parameters, i.e., pump power and SOD amount, it is found that the PSS experiences multiple bifurcations before entering into chaos. The imperfect balance of SOD and self-phase modulation (SPM) provides a new scenario of period-doubling bifurcation dynamics that PSS exhibits the dynamical energy redistribution between the main lobe and the oscillating tails. In particular, a critical state that the PSS evolves spontaneously between the period-doubling bifurcation and chaos has also been demonstrated. These findings will give new insights into the period-doubling bifurcation dynamics as well as chaos in dissipative optical systems.

Nonlinear Energy Transfer of a Spar-Floater System Using the Inerter Pendulum Vibration Absorber

Abstract The inerter pendulum vibration absorber (IPVA) is integrated between a spar and an annulus floater using a ball-screw mechanism to study its wave energy conversion potential. Hydrodynamic stiffness, added mass, and radiation damping effects on the spar-floater system are characterized using the boundary element method. It is found that a 1:2 internal resonance via a period-doubling bifurcation in the system is responsible for nonlinear energy transfer between the spar-floater system and the pendulum vibration absorber. This nonlinear energy transfer occurs when the primary harmonic solution of the system becomes unstable due to the 1:2 internal resonance phenomenon. The focus of this paper is to analyze this 1:2 internal resonance phenomenon near the first natural frequency of the system. The IPVA system when integrated with the spar-floater system is shown to outperform a linear coupling between the spar and the floater both in terms of the response amplitude operator (RAO) of the spar and one measure of the energy conversion potential of the system. Finally, experiments are performed on the IPVA system integrated with single-degree-of-freedom system (without any hydrodynamic effects) to observe the 1:2 internal resonance phenomenon and the nonlinear energy transfer between the primary mass and the pendulum vibration absorber. It is shown experimentally that the IPVA system outperforms a linear benchmark in terms of vibration suppression due to the energy transfer phenomenon.

Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

Feigenbaum scenario without parameters.

Typically, the period-doubling bifurcations exhibited by nonlinear dissipative systems are observed when varying systems' parameters. In contrast, the period-doubling bifurcations considered in the current research are induced by changing the initial conditions, whereas parameter values are fixed. Thus, the studied bifurcations can be classified as the period-doubling bifurcations without parameters. Moreover, we show a cascade of the period-doubling bifurcations without parameters, resulting in a transition to deterministic chaos. The explored effects are demonstrated by means of numerical modeling on an example of a modified Anishchenko-Astakhov self-oscillator where the ability to exhibit bifurcations without parameters is associated with the properties of a memristor. Finally, we compare the dynamics of the ideal-memristor-based oscillator with the behavior of a model taking into account the memristor forgetting effect.

Period-doubling bifurcation in surface radio-frequency trap: Transition to chaos through Feigenbaum scenario.

We have numerically investigated the dynamics of charged microparticles in a "five-wire" surface radio-frequency trap. The period-doubling bifurcation conditions have been shown to depend on the particle, the trap, and the alternating voltage parameters. For a comprehensive study of the dynamics chaotization through a cascade of period doubling, we have used Fourier analysis of a particle trajectory as well as the calculations of a non-trivial Lyapunov exponent map. We have demonstrated that the period-doubling bifurcation is consistent with a Feigenbaum scenario. A new approach to particle property determination can, thus, be based on observing a period-doubling bifurcation.

Analysis of Leptospirosis transmission dynamics with environmental effects and bifurcation using fractional-order derivative

Mathematical formulations are essential tool to show the dynamics that how various diseases spread in the community. Differential equations with fractional or integer order can be utilized to see the effect of the dynamics direct or indirect Leptospirosis transmission, which are analyzed with different aspects. A mathematical description and dynamical sketch of Leptospirosis with environmental effects have been studied as a result of the successful efforts of various writers. In this study, we analyzed the Leptospirosis model described using a nonlinear fractional-order differential equation that takes the environmental effects into consideration. The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the Leptospirosis system is verified and test the system with flip bifurcation. Also system is investigated for global stability using Lyapunov first and second derivative functions. The existence, boundedness and positivity of the Leptospirosis is checked, which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects according to their sub-compartments. Solutions for fractional order system are derived with the help of advanced tool fractal fractional operator for different fractional values. Simulation are carried out to see symptomatic as well as a asymptomatic effects of Leptospirosis in the world wide, also show the actual behavior of Leptospirosis which will be helpful to understand the outbreak of Leptospirosis with environmental effects as well as for future prediction and control strategies.

Dynamical analysis of a discrete-time plant–herbivore model

In this paper, a discrete-time model of a plant–herbivore system is qualitatively analyzed using difference equations to describe population dynamics over time. The goal is to examine how the model behaves under varying parameter values and initial conditions. Results reveal that the model exhibits diverse dynamical behaviors such as stable equilibria, period-doubling cascade, and chaotic attractors. The analysis indicates that changes in crucial parameters greatly affect the system’s dynamics. This study offers crucial insights into plant–herbivore systems and highlights the value of qualitative analysis in comprehending intricate ecological systems.

On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population

This research investigates the discrete prey–predator model by including harvesting on the predator population, in the sense of Caputo fractional derivative. We define the topological categories of the model fixed points. We demonstrate mathematically that, under certain parametric conditions, a fractional order prey-predator model undergoes both a Neimark–Sacker (NS) and a Period-doubling (PD) bifurcations. Using the central manifold and bifurcation theory, we present proof for NS and PD bifurcations. It has been discovered that the fractional order prey-predator model’s dynamical behavior is significantly influenced by the parameter values and the initial conditions. Two chaos control techniques have been used to eliminate the chaos in the model. In order to support our theoretical and analytical results and to illustrate complex and chaotic behavior, numerical simulations have been shown.

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

Does a New Electric Vehicle Manufacturer Have the Incentive for Battery Life Investment? A Study Based on the Game Framework

Motivated by the electric vehicle battery life performance, we studied the optimal investment decision-making behaviour of duopoly automakers. Based on the framework of game theory, this paper explores the influence of various parameters in the static game and dynamic game on the results, in combination with consumers’ preference for the battery life of electric vehicles. In the static game, a smaller investment coefficient is more beneficial to a firm that adopts an investment strategy rather than a firm that does not. When the investment coefficient increases, the difference between the two manufacturers will become smaller. The change of parameters in the dynamic game system may lead to complex dynamic phenomena, and the system will experience period-doubling bifurcation and N-S bifurcation from a stable state into a chaotic state. It will also significantly impact the basins of attraction, which affect the decision-makers’ initial choice. Consequently, we can use the control method to return the unstable system to stability. Based on these findings, some management insights and suggestions are presented.

Bifurcation analysis and chaos control of a discrete-time prey-predator model with Allee effect

In this study, a discrete-time prey-predator model based on the Allee effect is presented. We examine the parametric conditions for the local asymptotic stability of the fixed points of this model. Furthermore, with the use of the center manifold theorem and bifurcation theory, we analyze the existence and directions of period-doubling and Neimark-Sacker bifurcations. The plots of maximum Lyapunov exponents provide indications of complexity and chaotic behavior. The feedback control approach is presented to stabilize the unstable fixed point. Numerical simulations are performed to support the theoretical results.

Bifurcation analysis and optimal control of discrete SIR model for COVID-19

Our research focuses on studying a COVID-19 SIR model for discrete cases, incorporating bi-linear phenomena and a constant rate of recovery. We have interested in analyzing the conditions for the existence of disease-free equilibrium and local equilibrium in this model. Additionally, we are investigating the stability of several bifurcations, including saddle–node bifurcations, flip bifurcations, 1:1 resonances, and Neimark-Sacker bifurcations. These bifurcations provide insights into the behavior and dynamics of the system as specific parameters change. We propose a vaccine response rate to introduce a control variable for the SIR system. This variable represents the rate at which individuals receive vaccinations, which can affect the spread of the disease. Finally, demonstrate an epidemic model based on vaccination control with social awareness, using the results and insights obtained from our analysis to showcase the effectiveness of the proposed control strategy in mitigating the spread of COVID-19. In our research, we also employ Pontryagin’s maximum principle to solve the system. By applying this principle, we can determine the optimal control strategy that minimizes the spread of the disease or achieves other desired objectives. We have conducted a comprehensive analysis of the COVID-19 SIR model with a focus on bifurcations, control strategies, and social awareness about the impact of vaccination. Such research contributes to our understanding of the dynamics of infectious diseases and provides valuable insights for controlling epidemics.

Complex Dynamics Modes in a Simple Model of Prey-Predator Community: Bistability and Multistability

The paper proposes and studies a two-component discrete-time model of the prey-predator community considering zooplankton and fish interactions and their development features. Discrete-time systems of equations allow us to take into account naturally the rhythm of many processes occurring in marine and freshwater communities, which are subject to cyclical fluctuations due to the daily and seasonal cycle. We describe the dynamics of fish and zooplankton populations constituting the community by Ricker’s model, which is well-studied and widely used in population modeling. To consider the species interaction, we use the Holling-II type response function taking into account predator saturation. We carried out the study of the proposed model. The system is shown to have from one to three non-trivial equilibria, which gives the existence of the complete community. In addition to the saddle-node bifurcation, which generates bistability of stationary dynamics, a nontrivial equilibrium loses its stability according to the Neimark-Sacker scenario with an increase in the reproductive potential of both predator and prey species, as a result of which the community exhibits long-period oscillations similar to those observed in experiments. With the higher bifurcation parameter, the reverse Neimark-Sacker bifurcation is shown to occur followed by the closed invariant curve collapses, and dynamics of the population stabilizes, later losing stability through a cascade of period-doubling bifurcations. Multistability complicates the birth and disappearance of the invariant curve in the phase space scenario by the emergence of another irregular dynamics in the system with the single unstable nontrivial fixed point. At fixed values of the model parameters and different initial conditions, the system considered is shown to demonstrate various quasi-periodic oscillations. Despite extreme simplicity, the proposed discrete-time model of community dynamics demonstrates a wide variety and variability of dynamic modes. It shows that the influence of environmental conditions can change the type and nature of the observed dynamics.

The role of fluctuations in quantum and classical time crystals

Discrete time crystals (DTCs) are a many-body state of matter whose dynamics are slower than the forces acting on it. The same is true for classical systems with period-doubling bifurcations. Hence, the question naturally arises what differentiates classical from quantum DTCs. Here, we analyze a variant of the Bose-Hubbard model, which describes a plethora of physical phenomena and has both a classical and a quantum time-crystalline limit. Fluctuations enter the system due to the intrinsic quantum uncertainty and/or due to finite coupling to an environment. These fluctuations can activate transitions between the system’s various stationary solutions. We study the role of fluctuations on the stability of the system in the long-time limit and find no distinction between quantum and classical DTCs. This allows us to probe the fluctuations in an experiment using two strongly coupled parametric resonators subject to classical noise.

Quasi-periodic structure in chaotic bursting attractor for a controlled jerk oscillator

The main purpose of the paper is to show that parts of the trajectory for chaotic bursting oscillations may oscillate according to different limit cycles, which implies orderly movement exists in chaos. For a typical jerk oscillator with a cubic nonlinearity, when a slow-varying controlling term is introduced, bursting oscillations may occur, appearing in the combination of large-amplitude and small-amplitude oscillations. Two coexisting stable equilibrium points may evolve to chaos via different cascades of period-doubling bifurcation, respectively, which means different types of stable attractors may coexist with the variation of slow-varying parameter, while several periodic windows exist in the chaotic region. It is found that the spiking oscillations of the full system may change according to different cycles not only for 2-D torus bursting oscillations but also for the chaotic bursting attractor. Furthermore, whether the influence of the short periodic window of the fast subsystem on the bursting oscillations appears or not depends on the size of the slow scale, since the inertia of the movement may cause the delay effect of the bifurcations at both two edges of the window. The findings suggest that coupling of multiple scales may be one of key factors for the diversity of real world.

Instabilities near Ultrastrong Coupling in a Microwave Optomechanical Cavity.

With artificially engineered systems, it is now possible to realize the coherent interaction rate, which can become comparable to the mode frequencies, a regime known as ultrastrong coupling (USC). We experimentally realize a cavity-electromechanical device using a superconducting waveguide cavity and a mechanical resonator. In the presence of a strong pump, the mechanical-polaritons splitting can nearly reach 81% of the mechanical frequency, overwhelming all the dissipation rates. Approaching the USC limit, the steady-state response becomes unstable. We systematically measure the boundary of the unstable response while varying the pump parameters. The unstable dynamics display rich phases, such as self-induced oscillations, period-doubling bifurcation, and period-tripling oscillations, ultimately leading to the chaotic behavior. The experimental results and their theoretical modeling suggest the importance of residual nonlinear interaction terms in the weak-dissipative regime.